Numerical modeling of rock deformation: 03 Analytical methods - Folding. Stefan Schmalholz LEB D3

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1 Numerical modeling of rock deformation: 0 Analytical methods - Folding Stefan Schmalholz schmalholz@erdw.ethz.ch LEB D S008, Thursday 0-, CAB 5 Overview Derive a thin-plate equation for bending, folding and buckling Application of linear stability analysis Dominant wavelength concept for folding

2 Force balance in x-direction sin β = β, cos β = ( ) β ( )( β β) 0 force balance in the x-direction N + N + dn Q + Q + dq + d = dn + dqβ + Qdβ + dqdβ dn + dqβ + Qdβ / dx division by dx N Q β + β + Q introducing derivatives N small shear stress in slender beams drop quadratic terms including small shear stress Q and small rotation β Force balance in z-direction sin β = β, cos β = ( ) β ( )( β β) 0 force balance in the z-direction Q + Q + dq N + N + dn + d = dq + dnβ + Ndβ + dndβ dq + dnβ + Ndβ / dx division by dx Q N β β N introducing derivatives Q β N use result from force balance in the x-direction

3 Moment balance ( ) 0 moment balance M M + dm + Qdx = M + Q introducing derivatives Governing balance equations N Q β N M + Q Force balance x Force balance z Moment balance N β = M N x x 0

4 Bending moment and strain Stress in beam E E w σxx = ε xx y ν ν x Bending moment E M = yσ dy = y ε xxdy = ν xx E w E w ν ν y dy = y dy = = D E w w ( ν ) Bending equation w M = D w β = x N = P M β N x x w w D P 0 + =

5 Various bending equations Bending of elastic crust due to topography under gravity Buckling of elastic crust under gravity w x D + ( ρm ρc) gw= ρcghcsin π λ w w D P gw + + ( ρm ρc) Buckling of elastic layer in viscous media μ D w + P w + k w t Folding of viscous layer in viscous media 5 w με& w w μ k μ + + t t Application to lithosphere The deflection of the elastic oceanic lithosphere at an ocean trench can be described with a simple bending equation. Note that the so-called forebulge is a typical feature of elastic beam bending w D + ( ρm ρw) gw= 0 Turcotte & Schubert, Geodynamics 5

6 The dominant wavelength theory Natural single-layer folds seem to have a ratio of arc length to thickness which is relatively constant. Is there a mechanical explanation for this phenomenon? Does the geometry depend on the rheology? ow are these folds generated anyway? A = amplitude L = arc length λ = wavelength θ = max. limb dip Why do we care? ans Ramberg Maurice Biot Ray Fletcher Equation for viscous folding w w D P q t w με& w w μ k μ + + t t ε xx w σxx = μ μ y t t μ D = P = μ & ε w μ μ P= μ & ε w q= μk t λ = π k 6

7 Linear stability analysis This equations describes folding of a viscous layer embedded in viscous material subject to layer-parallel compression. 5 w με& w w μ k μ + + t t We want to see if a layer with small sinusoidal perturbations is stable under compression. Therefore, we investigate solutions of the form (, ) = exp( α ) cos ( ), (, ) = exp( α + ) w x t w t kx w x t w t ikx 0 0 which are periodic in space, x, but can grow or decay with time, t, depending on the sign of α. The layer is stable if α < 0 neutrally stable if α and unstable if α > 0 If α is complex we investigate the real part because the imaginary part represents periodic oscillations. k & k k Linear stability analysis After substituting the solution Ansatz into the governing equation we get μ α με + μ α The equation includes both the growth rate and the wave number. Solving for the growth rate α provides α μek = μ k + μ which is a dispersion relation, because it relates the growth rate of a perturbation to its wave number, or alternatively, wavelength. The maximum of α is found by setting the derivative of α with respect to k to zero and solving for k. Substituting the wavelength corresponding to the maximum back into the dispersion relation yields the maximal value of α. P 9P k μ π μ μ 0, λ π, α 6 α = = = = = k μ k + μ k μ μ ( μk + μ) α Dispersion curve λ 7

8 Numerical verification ~λ ~λ α μ = 6 μ α Dispersion curve Biot, 957 Ramberg, 96 λ = 6μ λ π μ Numerical verification 8